Dynamical Hall effect in a bounded two-dimensional classical plasma: edge-magnetoplasmon spectrum in the crossover regime and quantitative comparison for electrons on liquid helium

In 1985, Glattli, Andrei, Deville, Poitrenaud and Williams measured radio-frequency absorption of electrons confined to a disk on liquid helium in a perpendicular magnetic field and identified the dynamical Hall effect: the Lorentz force splits the clockwise/counter-clockwise azimuthal disk-magnetoplasmon modes, drives slow modes below the cyclotron frequency, and at high field produces a magnetically localized one-dimensional wave propagating along the plasma perimeter. Their theory, a quantization condition for the mode wave numbers, is valid only in two asymptotic regimes, and its idealized screened perimeter-wave asymptote $\nu c_p/R^*$ (azimuthal mode index $\nu$, screened plasmon velocity $c_p$, effective edge radius $R^*$) disagrees quantitatively with the measured asymptotic frequencies. Later work (Fetter's hydrodynamic edge-magnetoplasmon theory for half-plane and screened disk geometries; the Volkov-Mikhailov high-field theory for arbitrary edge profiles; experiments on helium in rings and channels) addresses only the asymptotic limits. A controlled, experimentally validated theory of the edge-mode spectrum through the crossover regime where the magnetic localization width $c_p/\omega_c$ is comparable to the edge-density decay length $D/\pi$ and screening is only partial does not exist, and the proposal to use magnetically localized plasmon waves as a probe of extended-state density in the quantum Hall regime remains unrealized.

Background

A two-dimensional (2D) plasma can be realized in an exceptionally clean, classical (non-degenerate) form as surface-state electrons trapped above liquid helium by the image potential (the electrostatic attraction between an electron and the polarization charge it induces in the helium) and by electrodes that close a cell of height $D$. The in-plane collective charge oscillations are plasmons; in a perpendicular magnetic field $H$ each electron executes cyclotron motion at the cyclotron frequency $\omega_c = eH/mc$, and the bulk magnetoplasmon dispersion of an infinite 2D layer screened by metallic plates a distance $D$ away is $\omega^2(k) = \omega_c^2 + (2\pi n_0 e^2/m)\,k\tanh(kD/2)$, where $n_0$ is the areal electron density and the factor $\tanh(kD/2)$ encodes finite-$kD$ screening (the interaction becomes effectively 3D Coulombic, screened at wavelength $\sim D$, at small $k$). In a finite sample the static Hall effect piles charge at the boundary, and the Lorentz force couples the transverse (Hall) current to density gradients.

Glattli, Andrei, Deville, Poitrenaud and Williams (Phys. Rev. Lett. 54, 1710 (1985)) demonstrated this dynamical Hall effect on a helium cell with $D = 2$ mm and disk radius $R = 9$ mm set by a guard-ring electrode: the degeneracy of clockwise and counter-clockwise azimuthal disk modes $(\pm\nu,\mu)$ (azimuthal index $\pm\nu$, radial index $\mu$) is lifted in small fields, and at high fields new modes appear whose frequencies decrease with field, drop below $\omega_c$, and become progressively localized in an edge strip: a magnetically localized one-dimensional wave propagating along the perimeter, the edge magnetoplasmon (observed the same month for an electron sheet on a helium film by Mast, Dahm and Fetter). The theory of the 1985 Letter, elaborated in the companion paper of Glattli, Andrei, Deville and Williams ('1D perimeter waves in a classical 2D electron system', Surface Science Letters 170, 70 (1986)), reduces the problem to an eigenvalue (quantization) condition for the in-plane wave numbers $k_{\pm\nu,\mu}$: a Bessel-function boundary equation at an effective radius $R^* = \int_0^{R-s} dr/\sigma(r)$, where $\sigma(r)$ is the static density profile, decaying over a characteristic edge length $\sim D/\pi$ near the edge recession $s$, with an explicit Hall term proportional to $\nu\omega_c/\omega$ and edge-profile corrections of order $D/\pi R$; it is valid to order $(D/\pi R)^2$ for $kD \lesssim 1$. In the step-profile limit (a density that is constant up to a sharp edge) the condition becomes $J_\nu'(kR_0) = -(\nu\omega_c/\omega)J_\nu(kR_0)$ (with $J_\nu$ a Bessel function, $R_0 = R-s$); once $\omega_c$ exceeds a threshold $c_p\sqrt{\nu(\nu+1)}/R_0$, where $c_p = \sqrt{\pi n_0 e^2 D/m}$ is the screened plasmon velocity, the wave number of the slow $(-\nu,1)$ mode turns imaginary, its frequency approaches the perimeter-wave asymptote $\nu c_p/R_0$ (written $\nu c_p/R^*$ in the paper's discussion of the experimental comparison), and its radial profile decays over the magnetic localization width $\sim c_p/\omega_c$.

The source paper itself flags what its framework cannot do. First, the quantization condition relies on approximations that break down when $|(\omega_c-\omega)/\omega_c| \ll D/\pi R$ or when the localization width $c_p/\omega_c$ becomes comparable to the edge-density decay length $D/\pi$; in those regimes different asymptotics apply (the unscreened-interaction limit, or local solutions of the type obtained in parabolic-cylinder coordinates) and require further analysis. Second, the measured asymptotic perimeter frequencies show quantitative discrepancies with the idealized screened estimate $\nu c_p/R^*$, so a full treatment of screening, of realistic density profiles when the localization length is comparable to the profile decay length, and of finite-$kD$ effects is needed. Third, the framework has not been worked out for other finite geometries (rectangular samples, where earlier unexplained modes had been reported) nor for the solid phase, in which the electrons form a Wigner crystal (a quantum-crystalline state of strongly correlated electrons) coupled to helium-surface dimple and ripplon modes (ripplons are the quantized capillary waves of the liquid-helium surface) with a local mode frequency $\omega_0$ and transverse response. Fourth, the paper speculatively proposes magnetically localized plasmon waves as a probe of the extended-state density in the quantized Hall regime (the regime of the integer quantum Hall effect, where a degenerate 2D electron gas has Landau levels and transport is carried by extended states near the Landau-level centers and by chiral edge states), which would require coupling the classical-fluid description to quantum localization physics and submicron edge-localization scales.

Since 1985, edge magnetoplasmons have become a standard collective excitation of confined 2D electron systems, studied on helium, on semiconductor heterostructures, and in graphene.

Problem Statement

Construct a quantitative theory of the collective-mode spectrum of a bounded two-dimensional classical plasma in a perpendicular magnetic field, valid across the whole field range from the disk magnetoplasmons of the 1985 framework to the high-field edge magnetoplasmon, and validate it against experiment on the electrons-on-helium disk. In detail, relative to the framework of Glattli et al. (1985), the open questions are: (1) extend the mode quantization theory (the eigenvalue condition for the wave numbers $k_{\pm\nu,\mu}$, the edge-mode threshold, and the perimeter-wave asymptote) beyond its two asymptotic regimes, so that it remains controlled in the crossover regime where the magnetic localization width $c_p/\omega_c$ becomes comparable to the characteristic edge-density decay length $D/\pi$ (and where $|(\omega_c - \omega)/\omega_c| \ll D/\pi R$), bridging the screened step-profile asymptotics and the unscreened-interaction limit, including finite-$kD$ screening and the realistic smooth edge profile $\sigma(r)$; (2) reconcile the experimentally observed asymptotic perimeter-mode frequencies, which deviate quantitatively from the idealized screened estimate $\nu c_p/R^*$, by a treatment of screening, realistic density profiles, and finite-$kD$ effects that is not idealized; (3) work out the same framework for other finite geometries (rectangular samples and channels) and for the solid (Wigner-crystal) phase in which local dimple modes of frequency $\omega_0$ and transverse excitations enter; and (4) establish or refute the paper's speculative proposal to use magnetically localized plasmon waves as a local probe of the extended-state density in the quantum Hall regime, which requires coupling the classical-fluid description to quantum-localization physics and demonstrating the probe at the submicron edge-localization scales envisaged. The aim is a controlled derivation of the spectrum (frequencies, splittings, and localization lengths) with quantitative agreement with measured spectra on electrons on helium, together with an assessment of the quantum-Hall probe proposal.

References

1. D. C. Glattli, E. Y. Andrei, G. Deville, J. Poitrenaud, and F. I. B. Williams, 'Dynamical Hall Effect in a Two-Dimensional Classical Plasma', Physical Review Letters 54, 1710-1713 (1985), DOI 10.1103/PhysRevLett.54.1710, https://doi.org/10.1103/PhysRevLett.54.1710
2. D. C. Glattli, E. Y. Andrei, G. Deville, and F. I. B. Williams, '1D perimeter waves in a classical 2D electron system', Surface Science Letters 170, 70 (1986), DOI 10.1016/0167-2584(86)90530-X, https://doi.org/10.1016/0167-2584(86)90530-X
3. D. B. Mast, A. J. Dahm, and A. L. Fetter, 'Observation of Bulk and Edge Magnetoplasmons in a Two-Dimensional Electron Fluid', Physical Review Letters 54, 1706-1709 (1985), DOI 10.1103/PhysRevLett.54.1706, https://doi.org/10.1103/PhysRevLett.54.1706
4. A. L. Fetter, 'Edge magnetoplasmons in a bounded two-dimensional electron fluid', Physical Review B 32, 7676-7684 (1985), DOI 10.1103/PhysRevB.32.7676, https://doi.org/10.1103/PhysRevB.32.7676
5. A. L. Fetter, 'Edge magnetoplasmons in a two-dimensional electron fluid confined to a half-plane', Physical Review B 33, 3717-3723 (1986), DOI 10.1103/PhysRevB.33.3717, https://doi.org/10.1103/PhysRevB.33.3717
6. A. L. Fetter, 'Magnetoplasmons in a two-dimensional electron fluid: Disk geometry', Physical Review B 33, 5221-5227 (1986), DOI 10.1103/PhysRevB.33.5221, https://doi.org/10.1103/PhysRevB.33.5221
7. V. A. Volkov and S. A. Mikhailov, Zh. Eksp. Teor. Fiz. 94, 217 (1988) [Sov. Phys. JETP 67, 1639 (1988)], the general high-field edge-magnetoplasmon theory
8. S. A. Mikhailov and V. A. Volkov, 'Inter-edge magnetoplasmons in inhomogeneous two-dimensional electron systems', Journal of Physics: Condensed Matter 4, 6523-6538 (1992), DOI 10.1088/0953-8984/4/31/005, https://doi.org/10.1088/0953-8984/4/31/005
9. P. J. M. Peters, P. Scheuzger, M. J. Lea, W. P. N. M. Jacobs, and R. W. van der Heijden, 'The AC Hall effect for electrons on liquid helium in ultra-high magnetic fields', Surface Science 305, 674-677 (1994), DOI 10.1016/0039-6028(94)90976-8, https://doi.org/10.1016/0039-6028(94)90976-8
10. P. K. H. Sommerfeld, P. J. M. Peters, H. F. W. J. Vorstenbosch, R. W. van der Heijden, A. T. A. M. de Waele, and M. J. Lea, 'Observation of harmonic, high-field, low-frequency edge-magnetoplasma resonances for electrons on helium', Physica B 194-196, 1311-1312 (1994), DOI 10.1016/0921-4526(94)90985-7, https://doi.org/10.1016/0921-4526(94)90985-7
11. P. J. M. Peters, P. K. H. Sommerfeld, S. van den Berg, P. P. Steijaert, R. W. van der Heijden, and A. T. A. M. de Waele, 'Low-frequency edge magnetoplasmons in a ring-shaped two-dimensional electron gas on liquid helium', Physica B 204, 105-110 (1995), DOI 10.1016/0921-4526(94)00249-U, https://doi.org/10.1016/0921-4526(94)00249-U
12. O. I. Kirichek, I. B. Berkutov, Yu. Z. Kovdrya, and V. N. Grigor'ev, 'Evidence for the edge magnetoplasmon acoustic mode in an electron layer over liquid helium', Journal of Low Temperature Physics 109, 397-405 (1997), DOI 10.1007/s10909-005-0092-2, https://doi.org/10.1007/s10909-005-0092-2
13. M. I. Goksu, M. Kim, K. A. Mantey, and A. J. Dahm, 'Edge magnetoplasmons in a partially screened two-dimensional electron gas on a helium surface', Physical Review B 75, 115323 (2007), DOI 10.1103/PhysRevB.75.115323, arXiv:cond-mat/0703558, https://doi.org/10.1103/PhysRevB.75.115323
14. S. Yamanaka, T. Arai, A. Sawada, A. Fukuda, and H. Yayama, 'Incompressible edge wave in classical two-dimensional electron liquid on helium surface', EPL 100, 17009 (2012), DOI 10.1209/0295-5075/100/17009, https://doi.org/10.1209/0295-5075/100/17009
15. S. Yamanaka, T. Arai, A. Sawada, A. Fukuda, and H. Yayama, 'Oscillation mode transformation of edge magnetoplasmons in two-dimensional electron system on liquid-helium surface', Low Temperature Physics 39, 842-850 (2013), DOI 10.1063/1.4823489, https://doi.org/10.1063/1.4823489
16. V. K. Talyanskii, M. Wassermeier, A. Wixforth, J. Oshinowo, J. P. Kotthaus, I. E. Batov, G. Weimann, H. Nickel, and W. Schlapp, 'Edge magnetoplasmons in the quantum Hall effect regime', Surface Science 229, 40-42 (1990), DOI 10.1016/0039-6028(90)90827-U, https://doi.org/10.1016/0039-6028(90)90827-U
17. I. M. Grodnensky, D. Heitmann, and K. von Klitzing, 'Single-particle and many-body effects in the edge magnetoplasma excitations of a two-dimensional electron system', Surface Science 263, 467-470 (1992), DOI 10.1016/0039-6028(92)90390-R, https://doi.org/10.1016/0039-6028(92)90390-R
18. M. N. Khannanov, A. A. Fortunatov, and I. V. Kukushkin, 'Manifestations of incompressible strips in the edge magnetoplasmon spectrum in the quantum Hall effect regime', JETP Letters 93, 657-660 (2011), DOI 10.1134/S0021364011110051, https://doi.org/10.1134/S0021364011110051
19. V. M. Muravev, A. M. Zarezin, P. A. Gusikhin, and I. V. Kukushkin, 'In What Magnetic Field Does an Edge Magnetoplasmon Arise?', JETP Letters 118, 439-442 (2023), DOI 10.1134/S0021364023602580, https://doi.org/10.1134/S0021364023602580
20. A. O. Badrutdinov, L. V. Abdurakhimov, and D. Konstantinov, 'Collective Excitations in a Two-Dimensional Electron System on Liquid Helium-4 under Cyclotron Resonant Condition', Journal of the Physical Society of Japan 84, 053601 (2015), DOI 10.7566/JPSJ.84.053601, https://doi.org/10.7566/JPSJ.84.053601
21. A. D. Chepelianskii, D. Papoular, D. Konstantinov, H. Bouchiat, and K. Kono, 'Coupled pair of one- and two-dimensional magnetoplasmons on electrons on helium', Physical Review B 103, 075420 (2021), DOI 10.1103/PhysRevB.103.075420, arXiv:1910.08026, https://doi.org/10.1103/PhysRevB.103.075420
22. I. Kostylev, M. Hatifi, D. Konstantinov, and A. D. Chepelianskii, 'Delocalized low-frequency magnetoplasmon in a two-dimensional electron fluid with cylindrical symmetry', arXiv:2404.07582 (2024), https://arxiv.org/abs/2404.07582
23. C. A. Mikolas, N. R. Beysengulov, A. J. Schleusner, D. G. Rees, C. Undershute, and J. Pollanen, 'Plasmon mode engineering with electrons on helium', Nature Communications 16 (2025), DOI 10.1038/s41467-025-60305-3, arXiv:2408.11228, https://doi.org/10.1038/s41467-025-60305-3
24. A. Kiiamov and D. Tayurskii, 'Magnetoplasmon engineering in microchannel-confined electrons on helium: edge modes, derivation of spectra, and quantitative coherence analysis', Journal of Low Temperature Physics 222 (2026), DOI 10.1007/s10909-026-03401-y, https://doi.org/10.1007/s10909-026-03401-y